Idea

Since a complex number can be understood as a pair of real numbers, this would naively reduce to analysis of pairs of functions of an even number of real variables; however by complex analysis we mean mathematical analysis which takes into account limits and derivative which do not depend on the real line in a complex plane on which we approach a point. This leads to the notions of holomorphic function, meromorphic function, etc. which are the main subject of complex analysis. However, there are connections to such real-analytic notions as harmonic analysis, and the geometric approach to complex analysis builds on the theory of smooth functions.